An solvable three-body model in finite volume

TL;DR

This paper introduces a novel method for solving the finite volume three-body problem by transforming it into a higher-dimensional two-body problem, demonstrated with a 1D three-particle scattering example.

Contribution

It presents a new approach that maps a three-body problem into a two-dimensional two-body problem, enabling solvability in finite volume settings.

Findings

Derived quantization condition for 1D three-body problem

Mapped 1D three-body problem to a 2D two-body problem

Demonstrated approach with a specific scattering example

Abstract

In this work, based on consideration of periodicity and asymptotic forms of wave function, we propose a novel approach to the solution of finite volume three-body problem by mapping a three-body problem into a higher dimensional two-body problem. The idea is demonstrated by an example of two light spinless particles and one heavy particle scattering in one spatial dimension. This 1D three-body problem resembles a two-body problem in two spatial dimensions mathematically, and quantization condition of 1D three-body problem is thus derived accordingly.